## Enumerations of maximum partial triple systems on 16 and 17 points

26 Oct 2017  ·  Demirkale Fatih, Donovan Diane, Grannell Mike ·

For $v\equiv 1$ or 3 (mod 6), maximum partial triple systems on $v$ points are Steiner triple systems, STS($v$)s. The 80 non-isomorphic STS(15)s were first enumerated around 100 years ago, but the next case for Steiner triple systems was unresolved until around 2004 when it was established that there are precisely 11084874829 non-isomorphic STS(19)s. In this paper we complete enumeration of non-isomorphic maximum partial triple systems for $v\le 19$. It is shown that there are 35810097 systems on 17 points and 47744568 on 16 points... We also establish that there are precisely 157151 non-isomorphic pairwise balanced designs, PBD($17,\{3,5\}$)s, having a single block of size 5. Structural properties of these systems are determined, including their automorphism groups, and the numbers of Pasch configurations, mitres and Fano planes contained in them. The systems themselves are available from the authors. read more

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Combinatorics